Work to line the top of an escape shaft with metal tubing is expected to finish in the next few hours at the mine where 33 men are trapped in northern Chile.
A winch and pulley will be installed and the shaft tested before the rescue begins, it is hoped, on Wednesday.
Amid the final preparations for the rescue the Chilean health minister said some of the miners were gallantly insisting they should be last to leave.
The men have been entombed since the mine partially collapsed on 5 August.
Among their relatives at the surface, initial despair at the accident gave way to anxiety over their predicament, to be replaced by joy and excitement as the hoped-for rescue approaches.
After 41 days of drilling and three separate rescue plans, engineers finally broke through to the miners on Saturday morning, sparking celebrations across Chile.
'Comradeship'
Engineers expect to finish lining the top 100m (330ft) of the escape shaft with metal tubing by about 0900 Monday (1200 GMT), said Mining Minister Laurence Golborne.
This will prevent rocks in the looser soil near the top of the shaft being dislodged and jamming the rescue capsule, named Phoenix.
They will then install the winch and pulley, before going on to test the function of the capsule.
Above the ground, the helicopter flights which will deliver the men to hospital as they emerge are being rehearsed a final few times.
The first four men to leave the mine have been selected by authorities, reports Chilean newspaper La Tercera. They have not been named but are thought to include some of the most psychologically stable and experienced of the miners - in case something goes wrong during the first few rescues.
Health Minister Jaime Manalich said some of the miners had shown a gallant spirit when he had spoken to them about the need to establish the order in which the men would leave.
"What was their reaction, the reaction of several of them? 'Yes minister, that's all good but I want to go last.' And then another one shows up, 'No, comrade, I had said that I would go last.' 'No, no, no. It's me,' would say another one.
"So I want to illustrate, through this anecdote from yesterday, that, in reality, they're maintaining a completely admirable spirit of solidarity, of comradeship," said the minister.
"There's no doubt that they will tell us later the details of this story from the entire time they were below, in the mine - that they've faced difficulties - but it's impressive how they themselves have managed things to maintain an enviable spirit which we all admire," he said.
